A349905 Arithmetic derivative of A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

0, 1, 1, 6, 1, 8, 1, 27, 10, 10, 1, 39, 1, 14, 12, 108, 1, 55, 1, 51, 16, 16, 1, 162, 14, 20, 75, 75, 1, 71, 1, 405, 18, 22, 18, 240, 1, 26, 22, 216, 1, 103, 1, 87, 95, 32, 1, 621, 22, 91, 24, 111, 1, 350, 20, 324, 28, 34, 1, 318, 1, 40, 135, 1458, 24, 119, 1, 123, 34, 131, 1, 945, 1, 44, 119, 147, 24, 151, 1, 837

Offset: 1

Author

Antti Karttunen, Dec 05 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003415, A003961, A026424 (positions of odd terms), A028260 (of even terms), A066829 (parity of a(n)).

Cf. also A343221, A343222, A347964, A347965, A348937, A353571.

Cf. A358760, A358761, A358762, A358763 for indices of terms that of the form 4k+j, for j=0..3, and A358750, A358751, A358752, A358753 for their characteristic functions.

Programs

Mathematica

f1[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := d[s[n]]; Array[a, 100] (* Amiram Eldar, Dec 05 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349905(n) = A003415(A003961(n));

Formula

a(n) = A003415(A003961(n)).